Signal-to-Noise Ratio⚠ unverified
Electrical / Signals · Signal-to-noise ratio (linear) from signal and noise power
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| signal_power | Ps | W | 100.0 | Signal power |
| noise_power | Pn | W | 1.0 | Noise power |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| snr | SNR | — | SNR (linear) |
The science & history
Understanding the Parameters
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$P_s$ — power in the wanted component (carrier, tone, or information-bearing waveform). Must be measured the same way as $P_n$ (same load resistance, same frequency band, same averaging).
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$P_n$ — noise power in that band: thermal ($kTB$), shot, amplifier NF, quantization, interferers, etc. Wider bandwidth captures more noise, so SNR depends on the noise bandwidth even when the signal is fixed.
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SNR — $P_s/P_n$. Example: $P_s = 100\,\text{W}$, $P_n = 1\,\text{W}$ → $\mathrm{SNR} = 100$ (often written 20 dB after the $10\log_{10}$ map). If powers are equal, SNR = 1 (0 dB).
Power vs amplitude. If you only have RMS voltages across the same resistance $R$, then $P \propto V_{rms}^2$, so $\mathrm{SNR} = (V_s/V_n)^2$. The linear calculator here takes powers directly.
Derivation (Approaching a Proof)
Noise and signal are treated as uncorrelated processes occupying a defined band. Their average powers add when measured on the same port:
$$P_{total} = P_s + P_n \quad\text{(uncorrelated)}.$$
The figure of merit that asks “how large is the signal relative to the noise?” is simply the ratio of those powers:
$$\mathrm{SNR} \equiv \frac{P_s}{P_n}.$$
This is a definition, not a circuit law. What is physical is how $P_s$ and $P_n$ arise: e.g. thermal noise $P_n = k_B T B$ into a matched load (Nyquist), signal power from Watt's Law $P = V I$ or $V_{rms}^2/R$, and degradations along a cascade (Friis noise-figure formula). The calculator evaluates only the ratio once both powers are known.
Why power (not amplitude) for the linear definition? Communication and information theory budgets are written in power (watts, dBm). Amplitude ratios are recovered via $\mathrm{SNR}_{voltage} = \sqrt{P_s/P_n}$ on a common $R$.
Relation to decibels
$$\mathrm{SNR_{dB}} = 10\log_{10}(\mathrm{SNR}) = 10\log_{10}(P_s/P_n).$$
Use SNR to Decibels for that conversion. The factor 10 (not 20) is required because SNR is already a power ratio; $20\log_{10}$ is for voltage/current ratios (see THD to Decibels).
History
SNR as a design metric grew with telephony and radio (early 20th century): how weak a signal could be and still be intelligible above receiver noise. Shannon’s channel capacity $C = B\log_2(1+\mathrm{SNR})$ (1948) made the linear power ratio a fundamental limit on reliable communication rate, not just an audio “clarity” number.
Related Concepts: SNR to Decibels, THD to Decibels, Total Harmonic Distortion, Watt's Law, Power Factor
Notes: Registry calculator signal-to-noise-ratio (unverified). Requires consistent bandwidth and
impedance for $P_s$ and $P_n$. Does not compute noise figure or $kTB$ from first principles.